Transition to turbulence

From CFD-Wiki

The process by which a flow initially laminar becomes turbulent is called transition. Transition is not a discontinuity, rather it involves a mechanism by which the flow changes from laminar through a state of intermittent laminar-turbulence and finally becomes fully turbulent. The distance and time over which this takes place depends on the particular flow.

It was established by Rayleigh in 1880 and later confirmed by Reynolds that transition is a stability problem. The flow can be seen as a dynamic system subject to perturbations. Viscous forces lead to the decay of these disturbances. When the Reynolds number increases beyond a critical limit, these disturbances can no longer be attenuated and stability is lost. In a sense we could say laminar flow and turbulence flows are special solutions of the navier stokes equations. The latter predominates in flows with vanishing viscosity. The Reynolds number, defined as the ratio of the inertia to the viscous forces, can be also be understood as a dimensionless reciprocal of viscosity.

The mechanism by which disturbance waves are generated in shear and boundary layers from the upstream disturbances is termed receptivity (Morkovin). depending on their method of amplification, disturbances can be classified as spatial, temporal or spatio-temporal.

Fundamental studies of transition have been carried out using the linear stability theory developped by Rayleigh, Orr and Sommerfeld. This theory assumes a parallel flow and a superposition of 2D disturbances. It is well known among fluid dynamicists as the Orr-Sommerfeld equation. Given the amplitude and frequency(ies) of disturbances,a flow field can be analysed to see the evolution of these disturbances in space and time. Transition occurs where these are strongly amplified.

Over the years, engineering tasks have compelled researchers to consider alternatives to the rigourous and linear theory. Other methods have evoled:

• The e_n method based on correlations and partial application of the linear stability theory
• Empirical correlations with the help of which the critical Reynolds number is determined.

The advance of CFD and the importance of transition in the design of turbomachines or flight vehicles have raised the question of coupling transition correlations to CFD solvers.

...to be continued

References

• 1. Schlichting, H., Gaster, K. Grenschicht Theorie, 9th ed. Springer Verlag, 1996
• 2. Reed,H., Saric, W.,Hassan, H.AIAA Stability and Transition Short Course, San Francisco, June 3, 2006.
• 3. Mack, L.M. On the Application of Linear Stability theory to the Problem of the Supersonic Boundary layer Transition, AIAA Papaer No. 74-134.
• 4. Warren, E.W., Hassan, H.A. Alternative to the e_n Method for determining onset of transition, AIAA Journal, vol. 36, No.1, 1998.
• 5. B. Akih Kumgeh. Numerical Simulation of Hypersonic Boundary Layer Transition for Scramjet Inlet Applications using the Langtry Menter Transition Model. Master thesis, RWTH Aachen, 2007.
• 6. Menter, F.R., Langtry, R.B.,Likki, S.R., Suzen, Y.B.,Huang, P.G. A correlation based transition model using local variables, Part 1-Model Formulation, ASME GT2004-53452.
• 7. Reshotko, E. Progress, Accomplishment and Issues in Transition Research, 28th Fluid Dynamics Conference, AIAA 97-1815